Question

discrete probability distributions worksheet

1. you flip four coins. let x, the random variable, be the number of heads on all four coins.

a. list the sample space for the experiment.
b. what are the possible values for x?
c. is the random variable, x, continuous or discrete?
d. construct a probability distribution for this experiment.
x
p(x)
e. construct a histogram for the probability distribution in the space below.

Explanation:

Step1: List the sample space

Each coin has 2 outcomes (H - head, T - tail). For 4 coins, the sample space \(S\) is all possible combinations.
\(S=\{HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH,\THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT\}\)

Step2: Determine possible values of \(X\)

\(X\) is the number of heads. The number of heads can be \(0\) (e.g., \(TTTT\)), \(1\) (e.g., \(HTTT\)), \(2\) (e.g., \(HHTT\)), \(3\) (e.g., \(HHHT\)), \(4\) (e.g., \(HHHH\)). So \(X = 0,1,2,3,4\)

Step3: Classify the random variable

A discrete random variable has a countable number of possible values. Since \(X\) can take on 5 distinct (countable) values (\(0,1,2,3,4\)), \(X\) is discrete.

Step4: Construct the probability distribution

The probability of getting \(k\) heads in \(n = 4\) coin - flips is given by the binomial probability formula \(P(X=k)=\binom{n}{k}p^{k}(1 - p)^{n - k}\), where \(n = 4\), \(p=\frac{1}{2}\), and \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\)

• For \(k = 0\): \(\binom{4}{0}(\frac{1}{2})^{0}(1-\frac{1}{2})^{4}=\frac{4!}{0!(4 - 0)!}\times1\times(\frac{1}{2})^{4}=\frac{1}{16}\)
• For \(k = 1\): \(\binom{4}{1}(\frac{1}{2})^{1}(1-\frac{1}{2})^{3}=\frac{4!}{1!(4 - 1)!}\times\frac{1}{2}\times(\frac{1}{2})^{3}=\frac{4}{16}\)
• For \(k = 2\): \(\binom{4}{2}(\frac{1}{2})^{2}(1-\frac{1}{2})^{2}=\frac{4!}{2!(4 - 2)!}\times(\frac{1}{2})^{2}\times(\frac{1}{2})^{2}=\frac{6}{16}\)
• For \(k = 3\): \(\binom{4}{3}(\frac{1}{2})^{3}(1-\frac{1}{2})^{1}=\frac{4!}{3!(4 - 3)!}\times(\frac{1}{2})^{3}\times\frac{1}{2}=\frac{4}{16}\)
• For \(k = 4\): \(\binom{4}{4}(\frac{1}{2})^{4}(1-\frac{1}{2})^{0}=\frac{4!}{4!(4 - 4)!}\times(\frac{1}{2})^{4}\times1=\frac{1}{16}\)

The probability distribution table:

\(X\)	\(P(X)\)
\(1\)	\(\frac{4}{16}\)
\(2\)	\(\frac{6}{16}\)
\(3\)	\(\frac{4}{16}\)
\(4\)	\(\frac{1}{16}\)

Step5: Construct the histogram (description)

On the \(x\) - axis, mark the values \(X = 0,1,2,3,4\). On the \(y\) - axis, mark the probabilities. Draw bars such that the height of the bar for \(X = 0\) is \(\frac{1}{16}\), for \(X = 1\) is \(\frac{4}{16}\), for \(X = 2\) is \(\frac{6}{16}\), for \(X = 3\) is \(\frac{4}{16}\), and for \(X = 4\) is \(\frac{1}{16}\)

Answer:

a. \(S=\{HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH,\THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT\}\)
b. \(X = 0,1,2,3,4\)
c. Discrete
d.

\(X\)	\(P(X)\)
\(1\)	\(\frac{4}{16}\)
\(2\)	\(\frac{6}{16}\)
\(3\)	\(\frac{4}{16}\)
\(4\)	\(\frac{1}{16}\)

e. (Histogram description as above)